Dear Kingsford and others,
the term "unbiased" may mean different things in different disciplines.
Although OLS is "unbiased" in a statistical sense, it may still me
"wrong" (to avoid the term "biased" here) in an ecological sense.
Imagine spatial autocorrelation emerging from the omission of an
important variable which is in itself spatially autocorrelated (such as
soil moisture down a hill slope). A non-spatial OLS will be able to
calculate coefficient estimates for such a model, and the estimates will
be asymptotically unbiased, but ecologically the omission of an
important variable will introduce a tendency away from the underlying
"true" parameters. In such cases, spatial autocorrelation in the data
(due to a "wrongly" specified model) will "bias" parameter estimates -
but only in an ecological sense, not in a statistical! (Or, in other
words: "biased" in a statistical sense means something like incorrect
given the right model structure. However, in spatial ecology, the model
structure may be misspecified and hence the results are "away from the
truth".)
So, when I was (too nonchalently) writing of spatial models "doing away
with" spatial autocorrelation and hence to deliver "unbiased" estimates,
I was referring to the ecological "unbiasedness", not the statistical.
Kingsford Jones was absolutely correct to pick this point up!
I also agree with the point that the "trick" spatial models use to model
spatial autocorrelation may well contain very interesting ecological
information, say on the spatial scale of species interactions, foraging
behaviour, aggregational patterns and so forth. I thus particularly like
approach such as Spatial Mapping of Eigenvectors (e.g. Griffith &
Peres-Neto 2006) and Spatial Filtering (Diniz-Filho & Bini 2005) and
Spatial Wavelets (Carl et al. 2008), because their additional spatial
variables can be mapped and thus provide a geographical starting point
to elucidating the processes behind spatial autocorrelation.
Carsten
Carl, G., Dormann, C.F., & Kühn, I. (2008) A wavelet-based method to
remove spatial autocorrelation in the analysis of species distributional
data. Web Ecology, 8, 22-29.
Diniz-Filho, J.A. & Bini, L.M. (2005) Modelling geographical patterns in
species richness using eigenvector-based spatial filters. Global Ecology
& Biogeography, 14, 177-185.
Griffith, D.A. & Peres-Neto, P.R. (2006) Spatial modeling in ecology:
the flexibility of eigenfunction spatial analyses in exploiting relative
location information. Ecology, 87, 2603-2613.
Kingsford Jones wrote:
> On Mon, Nov 17, 2008 at 12:48 AM, Carsten Dormann
> <carsten.dormann at ufz.de> wrote:
>>> Dear Camilo,
>>>> I hope I interpret correctly what you want.
>> In AN(C)OVA you are primary interested to see, whether a variable
>> significantly contributes to the explanation of the observed variance,
>> right? Spatial models by and large try to "do away with" spatial
>> autocorrelation (SAC), so that coefficient estimates are unbiased by SAC.
>>>> Just wanted to point out that (as I assume Carsten is aware) the OLS
> estimates of model coefficients remain unbiased under spatial
> autocorrelation (or any other process resulting in non-zero values
> off-diagonal elements of the error covariance matrix). However, the
> GLS estimates ((X'\Omega^{-1}X)^{-1}X'\Omega^{-1}y, where y ~ (X\beta,
> \sigma^2 \Omega)) are BLUE by the Gauss-Markov Theorem.
>> Also, although I agree with Carsten that spatial modelers often try to
> 'do away with' spatial autocorrelation in an effort to get better
> estimates of the coefficients, I think this is often the wrong view.
> E.,g, in the GLS setting it is not uncommon for the spatial
> autocorrelation parameters to be of at least as much biological
> interest as the \betas.
>> Kingsford Jones
>>>>>> Hence, an applying the anova-function to, say, a spatial eigenvector mapping
>> GLM (function ME in spdep) will give you the explained deviance for each
>> effect, including the spatial eigenvectors.
>>>> ANCOVA and regression models are fundamentally identical, only they focus on
>> different aspects of the results (deviance explaind vs. coefficient
>> estimates). Spatial models are similar to mixed effect models (and sometimes
>> ARE mixed effect models), so I can see no reason why not to treat them in
>> the same way as any other regression/ANOVA-model: run a GLM, use anova(.,
>> test="Chisq") on the model, done.
>>>> Not all spatial methods may offer a generic anova-function, but the majority
>> does (gls in nlme does, glmmPQL can be (wrongly!) forced to respond by using
>> anova.lme(.), while spautolm and spsarlm provide no anova-function). In
>> these cases, you have to have to resort to model comparison, i.e. comparing
>> a spatial model with and without the effect of interest (obeying marginality
>> and nestedness of models). The difference in deviance explained can be
>> attributed to the effect of the omitted variable.
>>>> HTH,
>>>> Carsten
>>>> P.S.: Let me advertise some own work here, if I may (open access pdf on the
>> journal's or my homepage):
>> Dormann, C. F., J. M. McPherson, M. B. Araújo, R. Bivand, J. Bolliger, G.
>> Carl, R. Davis, A. Hirzel, W. Jetz, W. D. Kissling, I. Kühn, R. Ohlemüller,
>> P. R. Peres-Neto, B. Reineking, B. Schröder, F. M. Schurr, and R. Wilson.
>> 2007. Methods to account for spatial autocorrelation in the analysis of
>> species distributional data: a review. Ecography 30:609-628.
>> With R-code for all methods in the appendix, of course.
>>>>>> Camilo Mora wrote:
>>>>> Hi:
>>>>>> Does anyone know if it is possible to run an ANCOVA in R while accounting
>>> or
>>> controlling for spatial autocorrelation? I have found usefull information
>>> into
>>> how to account for spatial autocorrelaion in regression models but not
>>> much
>>> into how to deal with the problem in an ANCOVA.
>>>>>> Thanks,
>>>>>> Camilo
>>>>>> Camilo Mora, Ph.D.
>>> SCRIPPS Institute of Oceanography
>>> University of California San Diego
>>> San Diego, USA
>>> Phone: (858) 822 1642
>>>http://cmbc.ucsd.edu/People/Faculty_and_Researchers/mora/>>> And
>>> Department of Biology
>>> Dalhouisie University
>>> Halifax, Canada
>>> Phone: (902) 494 3910
>>>http://as01.ucis.dal.ca/fmap/people.php?pid=53>>>>>> _______________________________________________
>>> R-sig-ecology mailing list
>>>R-sig-ecology at r-project.org>>>https://stat.ethz.ch/mailman/listinfo/r-sig-ecology>>>>>>>>>>> --
>> Dr. Carsten F. Dormann
>> Department of Computational Landscape Ecology
>> Helmholtz Centre for Environmental Research UFZ Permoserstr. 15
>> 04318 Leipzig
>> Germany
>>>> Tel: ++49(0)341 2351946
>> Fax: ++49(0)341 2351939
>> Email: carsten.dormann at ufz.de>> internet: http://www.ufz.de/index.php?de=4205>>>> _______________________________________________
>> R-sig-ecology mailing list
>>R-sig-ecology at r-project.org>>https://stat.ethz.ch/mailman/listinfo/r-sig-ecology>>>>>> _______________________________________________
> R-sig-ecology mailing list
>R-sig-ecology at r-project.org>https://stat.ethz.ch/mailman/listinfo/r-sig-ecology>>
--
Dr. Carsten F. Dormann
Department of Computational Landscape Ecology
Helmholtz Centre for Environmental Research UFZ
Permoserstr. 15
04318 Leipzig
Germany
Tel: ++49(0)341 2351946
Fax: ++49(0)341 2351939
Email: carsten.dormann at ufz.de
internet: http://www.ufz.de/index.php?de=4205